

International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii
May 25, 2015 14:55–15:20, Функциональные пространства, Moscow, Steklov Mathematical Institute of RAS






My Japanese book «Theory of Besov spaces, including a remark on the space $S'$ over $P$»
Y. Sawano^{} ^{} Tokyo Metropolitan University

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Abstract:
Let ${\mathcal S}'$ denote the set of all Schwartz distributions
and ${\mathcal P}$ the set of all polynomials.
If we define ${\mathcal S}_\infty$ to be the set of all $f \in {\mathcal S}$
such that
$\int_{{\mathbb R}^n}x^\alpha f(x) dx=0$
for all $\alpha$,
we can consider the dual space ${\mathcal S}_\infty'$.
We know that ${\mathcal S}_\infty'$ is isomorphic to ${\mathcal S}'/{\mathcal P}$ as
linear spaces. But it seems to me that this is true topologically.
In my Japanese book, I wrote a proof but I have commited the mistake.
But recently I modified the proof.
My result is as follows.
Theorem.
Equip ${\mathcal S}'$ and ${\mathcal S}'_\infty$ with the weak star topology.
Then the restriction mapping from ${\mathcal S}'$ to ${\mathcal S}_\infty'$ is open.
Materials:
abstract.pdf (84.9 Kb)
Language: English
References

S. Nakamura, T. Noi, Y. Sawano, “Generalized Morrey spaces and trace operator” (to appear)

Y. Sawano, Theory of Besov Spaces, NihonHyoronsha, 2011, 440 pp. (in Japanese)

